Axigluon Couplings in the Presence of Extra Color-Octet Spin-One Fields
aa r X i v : . [ h e p - ph ] A ug Axigluon Couplings in the Presen e of ExtraColor-O tet Spin-One FieldsAlfonso R. ZerwekhCentro de Estudios Subatómi os and Instituto de Físi aFa ultad de Cien iasUniversidad Austral de ChileCasilla 567, Valdivia, ChileAbstra tIn this paper, we study how the intera tion of the axigluon withquarks is modi(cid:28)ed when we introdu e new olor-o tet spin-one (cid:28)eldsin a hiral olor model. We show that in this ase the strength ofthis intera tion is not ompletely determined by the gauge symmetryany more and an be signi(cid:28) antly weaker than the one predi ted inthe original hiral olor model. In this way, we reinterpret the non-observability of the axigluon at the Tevatron, not as a limit on theaxigluon mass, but as a limit on the strength of the axigluon ouplingto quarks.1 Introdu tionIt is inherent to s ien e to look forward to new horizons and explore everypossible path in the hope to in rease our knowledge about Nature. That'swhy, despite the amazing phenomenologi al su ess of the Standard Model,we are always tempted to extend its parti le ontent or its symmetry andstudy the properties of the resulting model. Of ourse there are good the-oreti al reasons for doing it: we need to explain many mysteries present inthe foundations of the Standard Model. Perhaps the best known example isthe problem of naturalness in the Higgs se tor. The ne essity of understand-ing why we observe a so large hierar hy between the ele troweak s ale and,1or example, the Grand Uni(cid:28) ation s ale, have motivated the onstru tionof models as di(cid:27)erent as supersymmetry, Te hni olor or Little Higgs, amongothers. In all these theories, the hierar hy problem is solved at the pri eof signi(cid:28) antly extend the parti le and symmetry ontents proposed by theStandard Model and veri(cid:28)ed to the level of 1% at LEP.Another intriguing, but mu h less popular, question is why we observe a sodi(cid:27)erent stru ture for the weak and strong intera tions. The weak intera tionis profoundly hiral while the strong intera tion makes no di(cid:27)eren e betweenleft and right fermions. In this ontext, and with the aim of uni(cid:28) ation intheir eyes, Frampton and Glashow [1, 2℄ proposed many years ago a family ofmodels where the lo al gauge symmetry of the strong intera tion is supposedto be SU (3) L × SU (3) R at high energies. Of ourse at some large energys ale this symmetry must be broken to its diagonal group whi h is identi(cid:28)edwith the usual QCD group: SU (3) c . A general predi tion of this kind ofmodels is the existen e of a massive olor-o tet spin-one parti le usually alled axigluon be ause it has an axial oupling to quarks. The strength ofthis oupling is, as we will see, ompletely di tated by the gauge symmetryand, in the simplest, model is equal to the gluon-quark intera tion. Of oursethe presen e of su h a parti le would introdu e a large ontribution in thedijet produ tion ross se tion at hadron olliders. Unfortunately, su h asignal has not been observed and stringent limits have been pla ed on theaxigluon mass [3℄. Phenomenologi al onstrains have also been obtained fromtop-antitop events [4℄.In this paper, we study how the intera tion of the axigluon with quarksis modi(cid:28)ed when we introdu e new olor-o tet spin-one (cid:28)elds. We will showthat in this ase the strength of this intera tion is not ompletely determinedby the gauge symmetry any more and an be signi(cid:28) antly weaker than theone predi ted in the original hiral olor model. In this way we an reinterpretthe non-observability of the axigluon at the Tevatron not as a limit on theaxigluon mass, but as a limit on the strength of the axigluon oupling toquarks.2 Chiral Color and the AxigluonWe start by re alling the origin of the axigluon in hiral olor models. Of ourse, this revision will be s hemati and we will not dis uss important2oints su h as the hoise of anomaly free representations for fermions1. Wewill on entrate only in those aspe ts we judge important for the axigluonphenomenology.Let us, then, onsider the following Lagrangian: L = −
12 tr { G Lµν G µνL } −
12 tr { G Rµν G µνR } + f (cid:8) D µ U † D µ U (cid:9) (1)where G Lµν = ∂ µ l ν − ∂ ν l µ − ig L [ l µ , l ν ] G Rµν = ∂ µ r ν − ∂ ν r µ − ig R [ r µ , r ν ] D µ U = ∂ µ U − ig L l µ U + ig R U r µ D µ U † = ∂ µ U † − ig R r µ U † + ig L U † l µ and l µ ( r µ ) is the gauge (cid:28)eld of SU (3) L ( SU (3) R ). Of ourse, the (cid:28)rst twoterms are invariant under SU (3) L × SU (3) R transformations and the thirdterm is a nonlinear sigma model term we in lude to des ribe the breakingdown of the original lo al gauge symmetry to SU (3) c . Although it is straight-forward to develop the model onsidering di(cid:27)erent oupling onstants for SU (3) L and SU (3) R [6℄ we are going to limit ourselves to the ase where g L = g R = g . We do it in order to simplify our analysis and be ause this wasthe hoi e made by authors of the original hiral olor model motivated byuni(cid:28) ation arguments[1℄.In the unitary gauge ( U = 1 ), the third term of equation (1) gives riseto a nondiagonal mass matrix for the gauge bosons.When we diagonalize themass matrix, we found the following eigenvalues and eigenve tors: m G = 0 m A = gf √ G µ = 1 √ l µ + r µ ) (2) A µ = 1 √ l µ − r µ ) (3)1A detailed dis ussion of this topi s an be found in [1℄, [2℄ and [5℄3here G µ is the gluon and A µ is the axigluon.The intera tion Lagrangian for the quarks an be written as: L = 12 g ¯ ψl µ γ µ (1 − γ ) ψ + 12 g ¯ ψr µ γ µ (1 + γ ) ψ (4)Inverting equations (2) and (3) we an write the Lagrangian in term ofthe physi al (cid:28)elds and we obtain: L = g √ ψG µ γ µ ψ + g √ ψA µ γ µ γ ψ. (5)In this minimal model, the gluon-quark and axigluon-quark intera tionterms have the same oupling onstant g QCD ≡ g/ √ . In more general mod-els, where g L = g R , it is not true and g appears modulated by trigonometri fun tions of a mixing angle[6℄.3 Our ModelFollowing the ideas previously developed by the author in [7℄ and [8℄, wepropose to modify the model des ribed above by adding new spin-one (cid:28)elds, L µ and R µ , whi h transform like gauge (cid:28)elds under SU (3) L and SU (3) R respe tively with a hara teristi oupling onstant g ′ . The Lagrangian thatdes ribes the gauge se tor of the model, in luding the e(cid:27)e tive symmetrybreaking term, is L = −
12 tr { G Lµν G µνL } −
12 tr { G Rµν G µνR }−
12 tr { ρ Lµν ρ µνL } −
12 tr { ρ Rµν ρ µνR } + M g ′ tr n ( gl µ − g ′ L µ ) o + M g ′ tr n ( gr µ − g ′ R µ ) o + f (cid:8) D µ U † D µ U (cid:9) (6)where G Lµν , G Rµν and U are the same than in the previous se tion while ρ Lµν and ρ Rµν are de(cid:28)ned by ρ Lµν = ∂ µ L ν − ∂ ν L µ − ig ′ [ L µ , L ν ] ρ Rµν = ∂ µ R ν − ∂ ν R µ − ig ′ [ R µ , R ν ] g ′ ≫ g and we will write our results in the (cid:28)rst order in g/g ′ . The eigenvalues, thatis, the masses of the physi al states are: m G = 0 m A = gf √ m G ′ = Mm A ′ = M Thus, the physi al spe trum is omposed of a (exa tly) massless gluon, anaxigluon with the same mass (in this limit) than in the minimal model andtwo degenerate (at this level of approximation) heavy gluon and axigluon.Noti e that the new mass s ale M is not onstrained and we an safelysuppose that it is large enough to prevent the observation of the heavy states.The normalized mass eigenve tors an be written as: G µ = 1 √ l µ + 1 √ r µ + g √ g ′ L µ + g √ g ′ R µ A µ = − √ l µ + 1 √ r µ − g √ g ′ (cid:18) − m A M (cid:19) L µ + g √ g ′ (cid:18) − m A M (cid:19) R µ G ′ µ = g √ g ′ l µ + g √ g ′ r µ − √ L µ − √ R µ A ′ µ = − g √ g ′ (cid:18) − m A M (cid:19) − l µ + g √ g ′ (cid:18) − m A M (cid:19) − r µ ++ 1 √ L µ − √ R µ (7)Be ause we have now two (cid:28)elds that transform as gauge (cid:28)elds for ea hgroup ( l µ and L µ for SU (3) L and r µ and R µ for SU (3) R ) any ombinationof the form g (1 − k ) l µ + g ′ kL µ and g (1 − k ′ ) r µ + g ′ k ′ R µ , where k and k ′ arearbitrary onstants, an be used to onstru t ovariant derivatives[7℄. Thismeans that the Lagrangian des ribing the gauge intera tion of quarks an bewritten as: 5 = 12 g (1 − k ) ¯ ψl µ γ µ (1 − γ ) ψ + 12 g ′ k ¯ ψL µ γ µ (1 − γ ) ψ ++ 12 g (1 − k ′ ) ¯ ψr µ γ µ (1 + γ ) ψ + 12 g ′ k ′ ¯ ψR µ γ µ (1 + γ ) ψ (8)In prin iple, k and k ′ are independent parameters but, again, for simpli itywe will assume that k = k ′ . Using Lagrangian (8) and the de(cid:28)nition of thephysi al (cid:28)elds, we an obtain the terms of the Lagrangian that ouple thegluon and the axigluon to quarks L = g √ ψG µ γ µ ψ + g √ − χ ) ¯ ψA µ γ µ γ ψ (9)where χ is de(cid:28)ned as: χ ≡ m A M k (10)Noti e that, while the oupling of the gluon to the quarks is the usualone and is independent of k (be ause is prote ted by the SU (3) c gauge sym-metry), the oupling onstant of the axigluon has been modi(cid:28)ed and nowdepend on the free parameter k . Of ourse we an use this freedom to hoi ean adequated valued k in order to make the axigluon invisible at the Teva-tron.Let us brie(cid:29)y omment about to keep k and k ′ as independent parameters.In this ase, additionally to the axial oupling, a new ve tor oupling wouldappear between the axigluon and the quarks, proportional to ( k ′ − k ) .4 ResultsNow we want to estimate the values of χ allowed by experiments. It isevident that, in order to keep the axigluon invisible in dijet observations atthe Tevatron, the following inequality must be satis(cid:28)ed: (1 − χ ) σ A ≤ σ (11)where σ is the maximum value of the produ tion ross se tion of a reso-nan e de aying into dijets, allowed by experiment at 95% C.L. and σ A is the6 χ M A (GeV) Figure 1: Limits for free parameter χ de(cid:28)ned in equation (10). The regionbetween the ontinuous lines is allowed. ross se tion for the produ tion of an axigluon and its de ay into dijets inthe usual minimal model ( k = 0 ).We use Cal hep [9℄ in order to estimate σ A for √ s = 1 . TeV, and | y | ≤ (where y is the rapidity of the jet) for values of the axigluon mass inthe interval [260 , GeV. For σ , we use the values reported in [3℄.The results are summarized in (cid:28)gure 4. The ontinuous urves representthe limits on χ obtained using the equality in (11). The allowed region isthe one between the urves. For χ = 1 , the axigluon ompletely de ouplefrom quarks but, we an see, a deviation of χ from of about is allowedfor the whole mass range onsidered here. But for low masses a deviation aslarge as is still possible.5 Con lusionIn this work we have presented a new me hanism to redu e the oupling ofthe axigluon to quarks in order to evade the limits imposed by dijet pro-du tion experiments. This me hanism onsists in adding new spin-one (cid:28)eldsthat originally transform like gauge (cid:28)elds under SU (3) L and SU (3) R respe -tively and onstru t with them generalized ovariant derivatives. We want7o emphasize that our method is rather inno uous. Our model preserve thefundamental stru ture of hiral olor models and from equation (7) we ansee that the des omposition of the physi al states in terms of the originalhas the same stru ture for the light (cid:28)elds (the gluon and the axigluon) andthe heavy ones. That implies that their intera tion to quarks has also thesame stru ture. Thus, we expe t that on e one has hosen anomaly free rep-resentations for quarks in the minimal model, the potentially new anomalousdiagrams introdu ed in our model by the heavy states will also an el. Onthe other hand, although we have shown that our proposal works in the im-portant ase where we keep the symmetry between the left and right se tors,it is straightforward to generalize it to the ase when g L = g R . In this ase,we an redu e independently the axial as well as the ve tor oupling that theaxigluon develops in this generalized models. For doing that, it is enough tokeep k = k ′ in equation (8). Finally, the appearan e of a heavy gluon anda heavy axigluon is not problemati from a phenomenologi al point of viewsin e the model allows their mass to be large enough to be invisible at theTevatron.A knowledgementA. R. Z. is partially supported by Fonde yt grant 1070880. The author wantsto thank the hospitality of the Instituto de Físi a Teóri a (Unesp), São Paulo,Brazil, where this work was ompleted.TGDReferen es[1℄ P. H. Frampton and S. L. Glashow, (cid:16)Uni(cid:28)able Chiral Color With NaturalGim Me hanism,(cid:17) Phys. Rev. Lett. 58 (1987) 2168.[2℄ P. H. Frampton and S. L. Glashow, (cid:16)Chiral Color: An Alternative tothe Standard Model,(cid:17) Phys. Lett. B 190 (1987) 157.[3℄ T. Aaltonen et al. [CDF Collaboration℄, (cid:16)Sear h for new parti les de- aying into dijets in proton-antiproton ollisions at sqrt(s) = 1.96 TeV,(cid:17)Phys. Rev. D 79 (2009) 112002 [arXiv:0812.4036 [hep-ex℄℄.[4℄ P. Ferrario and G. Rodrigo, (cid:16)Constraining heavy olored resonan es fromtop-antitop quark events,(cid:17) arXiv:0906.5541 [hep-ph℄.85℄ P. H. Frampton, (cid:16)Gauge Field Theory(cid:17), John Wiley & Sons, INC, 2000,se ond edition.[6℄ M. V. Martynov and A. D. Smirnov, (cid:16)Chiral olor symmetry and possible G ′′